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As we discussed above, propositional meanings are modeled as context update functions in dynamic systems. As pointed out by Dotlaˇcil and Roelofsen (2019), context updates in non-inquisitive systems such as GSV and PCDRT can be divided into two classes: (a)constructiveupdates that introduce new discourse referents and create new possibili- ties, and (b)eliminativeupdates which remove possibilities. In an inquisitive setting such asInqD, the notion ofeliminativeupdates need to be revised. First, instead of possibilities,

Object Type Abbreviation

drefassignment function (re) -

drefassignment matrix ((re)t) m

Possibility (s×m) -

information state ((s×m)t) i

context (it) k

Table 3.1: Types and Abbreviation Conventions

such updates eliminateinformation states(while preserving the downward-closure); sec- ond, by eliminating information states, an update function doesn’t necessarily provide new information - it may also raise issues by carving out the alternatives that resolve it. In this section, we take a macro perspective on update functions, namely, we look at how context can be changed given an arbitrary proposition, without inspecting the inner structure of the proposition. A compositional fragment will be provided in the next section.

3.2.2.1 Informative and Inquisitive Context

Let’s first lay out some basic notions regarding the (inquisitive) properties of a contextc. The major properties of update functions, as we will see later, can then be characterized as their potential to change the properties of a context. The content of this part is simply an extension of the staticInqB to its dynamic counterpart, and all the notions defined

here can be traced back to the ones defined in §2.2.1.1. Therefore we will simply display the formal notions with minimal explanations.

Definition3.8. (Informative Content)

For any contextc, itsInformative Contentinfo(c) :=∪c. Definition3.9. (Informativeness)

A contextcwith domainδisInformativeiffthere is a possibility⟨w,G⟩with domain

δsuch that⟨w,G⟩<info(c). Otherwise,cisUninformative.

With the requirement of downward-closure, a context c, just like an inquisitive proposition, can be represented by means of itsmaximal elements, i.e. thealternatives. The

InqDversion of alternatives is defined as follows.

Definition3.10. (Alternatives)

The set ofAlternativesof a contextc,alt(c) :={s∈c|there is not∈csuch thatt⊃s}.

With the notion of alternatives, we can further define theinquisitivenessof a context. Definition3.11. (Inquisitiveness)

A contextcisInquisitiveiff|alt(c)|>1, or equivalently,info(c)<c.19

Finally, we define the notion of trivial contexts, the initial context and inconsistent context as follows.

19_{The equivalence, however, only holds when the set of possibilities is finite, which we will assume to}

Definition3.12. (Trivial Contexts)

A contextcistrivialiffit is neither informative nor inquisitive. Definition3.13. (The Initial Contextc_⊤)

Theinitial contextc_⊤is the trivial context whose domain is empty. Definition3.14. (The Inconsistent Contextc_⊥)

The inconsistent contextc_⊥ :={∅}.

3.2.2.2 ConstructiveUpdate: Context Extension and Subsistence

As mentioned above, context updates can function in aconstructive way and anelimi- nativeway. In static settings, on the other hand,drefinformation is left out, so context updates are usually reduced to theeliminativecases only. Whereas in dynamic settings, context updates also refer to cases where context is extended with newdrefs that are further encoded into thedomainof the context. In this section, the definition ofextensions

will be spelled out, along with a special case calledsubsistence (following Groenendijk et al., 1995). Note that the notion ofextensionsis compatible with eliminative operations (in fact it even rejects the introduction of contradictory information), but we believe it can be better understood in terms of domain extensions w.r.t.drefs. As a suitable illustration, therefore, we will introduce the very first update function inInq_D, namely [u], that stands for the introduction of adrefu.

In principle, anextensionof a contextcshould satisfy the following two conditions: (i) it maintains the world information established by c, and possibly add compatible pieces; and (ii) it maintains the discourse information established byc, and possibly add new ones. The second condition, in particular, is crucial for us to understandextensions

in a dynamic system. Therefore, here we first specify what it means to extend adref

assignment function/matrix:

Definition3.15. (ExtendingdrefAssignment Functions and Matrices)

(i) Adrefassignment function g′ is anExtensionof anotherdrefassignment function

g, written as g′ ≥ g, iffdom(g′)⊇dom(g), and for allu′ ∈ dom(g)\{u}, g(u)= g′(u). (ii) A drefassignment matrixG′ is an Extension of anotherdrefassignment function

G, written as G′ ≥ G, ifffor every g′ ∈ G′, there is g ∈ Gsuch that g′ ≥ gand for every g∈G, there is g′ ∈G′ such that g′ ≥ g.

That is, an extension of adrefassignment function/matrix does notdestroydiscourse information that is already established, only creates new ones. Based on this notion, we can further define the extensions of possibilities, information states, and finally contexts. Definition3.16. (Extending Possibilities)

A possibility⟨w′,G′⟩is anExtensionof another possibility⟨w,G⟩, written as⟨w′,G′⟩ ≥

⟨w,G⟩, iffw=w′ andG′ ≥G.

Definition3.17. (Extending Information States)

An information state s′ is an Extension of another information state s, written as

Definition3.18. (Extending Contexts)

A context c′ is an Extension of another context c, written as c′ ≥ c, iff for every information states′ ∈c, there iss∈ csuch thats′ ≥s.

Note that the Definition 3.17, 3.18 of extensions of information states and contexts differ in form from Definition 3.15 in that 3.17, 3.18 only require each member of the extended set to have a counterpart in the original set, but 3.15 requires each element in the original set to be extended as well. The difference corresponds directly to the fact that world information in the context can be eliminated. However, it is useful to also define a specific kind of context extension that only involves the addition of discourse information. Following Groenendijk et al. (1995), such extension is calledsubsistence. Definition3.19. (Subsistence of one Information States in another)

An information statessubsistsin another information states′, written ass≼s′, iffs′ ≥ sand for every possibility⟨w,G⟩∈s, there is⟨w′,G′⟩ ∈s′ such that⟨w′,G′⟩ ∈s′ ≥⟨w,G⟩.

Definition3.20. (Subsistence of an Information State in a Context)

An information statessubsistsin a contextc, written ass≼c, iffthere is one or more

s′ ∈csuch thats≼s′. Suchs′ is called adescendantofsinc. Definition3.21. (Subsistence of a Context in another)

A context c subsists in another contextc′, written asc ≼ c′, iff c′ ≥ c and for every

s∈c,s≼c′.

The subsistence of a context cin another one c′ can also be phrased asc support c′. Now, let’s get a bit more specific and introduce the first update function in Inq_D—the function [u]—that introduces adrefindexed byu. [u] is a context update function, thus of type (kk); and it can informally described as taking a contextc as input, and output a contextc′ such thatc ≼ c′ (no world information destroyed) andc′ is enriched fromc

with a newdrefu. To formally define it, we make use of the following sentences from the logical vocabulary:

(78) a. g[u]g′ :=[dom(g′)=dom(g)∪ {u}]∧ ∀v∈ dom(g)∩dom(g′) : g(v)= g(v′) where g,g′ aredrefassignment functions of type (re).

b. G[u]G′ := ∀g∈G: ∃g′ ∈G′.g[u]g′∧

∀g′ ∈G′ : ∃g∈G.g[u]g′

whereG,G′aredrefassignment matrices of typem. c. p[u]p′ :=π1(p)=π1(p′)∧π2(p)[u]π2(p′), where

- p,p′are possibilities of type (s×m);

- π1, π2 are projection function such that for any possibility p = ⟨w,G⟩,

π1(p) = w, π2(p) = G. In the following, we will also write π1(p) as wp,

π2(p) asGp.

The logical expressions given in (78) instantiate the informal description of the intro- duction ofuon different levels - fromdrefassignment function in (78a), todrefmatrix in (78b), and to possibility in (78c). Based on these notions, the semantic entry of [u] is given as follows:

As we can see, the condition corresponds to the definition of subsistence directly, and in addition, the extension is achieved through the introduction of u. Let’s illustrate with an example. Consider an input contextcwith only two entitesa,b, and an empty domain, as shown in Fig. (3.1a). An application of [u] on c then yields a new context that is extended from c with arbitrary sets of values for u, as shown in Fig. (3.1b). In the current and the rest of the diagrams, each black dot will represent a possibility. The world component is specified above it and fixed for each column, and thedrefmatrix is specified to its left and fixed for each row. The context will be represented by dashed rectangles enclosing its alternatives. Here the difference between wa,wb,wa,band w∅ is

neither specified nor made use of, but it will have effect when we introduce eliminative

updates. wa wa,b wb w∅ ∅ (a) wa wa,b wb w∅ u/a u/b u/a,u/b u/a⊕b u/a⊕b,u/a u/a⊕b,u/b u/a⊕b,u/a,u/b (b)

Figure 3.1: Application of thedrefintroduction operator [u]

3.2.2.3 EliminativeUpdate: Issues and Information

The last section explained how contexts can be updated/extendedconstructivelyvia in- troductions of discourse information. In this section, we turn to the more familiar kind of context updates, namelyeliminative updates. Meanwhile, as mentioned above, the notion ofeliminativeupdates has been enriched in the inquisitive setting, as such update may both provide information and raise issues. Below, we will provide formal notions that characterize the informativeness and inquisitiveness of an update function. More- over, two special relations that hold between a context and an update functions, namely

support and consistency, and a relation between update functions, namely entailment, will be provided. As promised, we will stay on a macro perspective and only reason about an arbitrary update functionAof type (kk). The various instantiations of update functions with different properties will be discussed in the next section, with natural language counterparts. In the following, we denote the context resulting from applying

Ato an input contextcasA(c).

An update function may not be well-defined for every contextc, that is, it may be a partial function. In particular, if an update functionArefers to adrefuthat is not in the domain (see Definition 3.7) a contextc, thenA(c) will be undefined. Now let’s proceed

to the definitions of inquisitiveness and informativeness of an update function. Definition3.22. (Informative and Inquisitive Update Functions)

• An update function A is informative iff there exists an uninformative context c

such thatA(c) is defined and informative.

• An update functionAisinquisitiveiffthere exists a non-inquisitive contextcsuch thatA(c) is defined and inquisitive.

Meanwhile, an update function can be contradictory or tautologic: Definition3.23. (Contradictions and Tautologies)

• An update functionAis aContradictionifffor any contextc,A(c)=c_⊥. • An update functionAis aTautologyifffor any contextc,A(c)=c.

We mentioned in the previous chapter that the notion ofsubsistencebetween contexts, sayc ≼ c′, can be rephrased asc supports c′. Here, we use the notion of subsistence to define the support relation between a context and an update function:

Definition3.24. (Support)

A contextcsupportsan update functionAiffA(c) is defined andc≼A(c).

Even if a context doesn’t support an update function, they can still be consistent: Definition3.25. (Consistency)

An update functionAisconsistentwith a contextciffA(c) is defined andA(c),c_⊥. Finally, we define the entailment relation for update functions. The notion of entail- ment in Dotlaˇcil and Roelofsen (2019) follows the “update-to-test” notion of entailment, following Groenendijk et al. (1995):

Definition3.26. (Entailment)

LetA1, ...,AnandBbe update functions. ThenA1, ...,AnentailB, written asA1, ...,An ⊨

B, ifffor every contextcsuch thatAn(...A1(c)) is defined,An(...A1(c)) supportsB.